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<div id="Sx1" class="ltx_section">
<h1 class="ltx_title ltx_title_section">Wavelet filter</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">The wavelet transform is commonly used in modern signal-processing
applications. This type of wavelet filter applies the <em class="ltx_emph">à trous</em>
algorithm <cite class="ltx_cite">[<a href="#bib.bib35" title="Astronomical Image and Data Analysis" class="ltx_ref">2</a>, <a href="#bib.bib16" title="Wavelet analysis for single molecule localization microscopy" class="ltx_ref">1</a>]</cite> which is an undecimated scheme
in which the filter responses are up-sampled, thereby inserting holes
(<em class="ltx_emph">trous</em> in French) between the filter coefficients. The output
of each filter level, therefore, contains the same number of samples
as the input.</p>
</div>
<div id="Sx1.p2" class="ltx_para">
<p class="ltx_p">The wavelet transform of the input data at level <img id="Sx1.p2.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi24.png" width="100" height="19" alt="j=1,\ldots,N"> is
computed as</p>
<table id="Sx1.E1" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E1.m1" class="ltx_Math" style="vertical-align:-7px" src="mi/mi10.png" width="129" height="21" alt="F_{j}=V_{j-1}-V_{j}\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(1)</span></td>
</tr>
</table>
<p class="ltx_p">where</p>
<table id="Sx1.E2" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E2.m1" class="ltx_Math" style="vertical-align:-9px" src="mi/mi11.png" width="177" height="27" alt="V_{j}=\left(V_{j-1}*\boldsymbol{k}_{j}\right)*\boldsymbol{k}_{j}^{\top}\,."></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(2)</span></td>
</tr>
</table>
<p class="ltx_p">Here <img id="Sx1.p2.m2" class="ltx_Math" style="vertical-align:-5px" src="mi/mi17.png" width="55" height="18" alt="V_{0}=I"> is the input image, <img id="Sx1.p2.m3" class="ltx_Math" style="vertical-align:-7px" src="mi/mi21.png" width="23" height="21" alt="\boldsymbol{k}_{j}"> is a filter
kernel (see <a href="Convolution.html" title="" class="ltx_ref">convolution with separable kernels</a>)
at the corresponding level, and <img id="Sx1.p2.m4" class="ltx_Math" style="vertical-align:-2px" src="mi/mi6.png" width="10" height="16" alt="l"> is the kernel size. The number
of levels is given by <img id="Sx1.p2.m5" class="ltx_Math" style="vertical-align:-2px" src="mi/mi16.png" width="20" height="16" alt="N">.</p>
</div>
<div id="Sx1.p3" class="ltx_para">
<p class="ltx_p">The expression in Equation (<a href="#Sx1.E1" title="(1) ‣ Wavelet filter" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>) acts as a band-pass
filter. Each wavelet level thus corresponds to a different range of
spatial frequencies. The first level <img id="Sx1.p3.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi14.png" width="23" height="18" alt="F_{1}"> contains mainly the
high spatial frequencies present in the input image including noise.
Higher levels contain ranges of lower and lower image frequency components.
In our case, the output of the wavelet filter is the second wavelet
level <img id="Sx1.p3.m2" class="ltx_Math" style="vertical-align:-5px" src="mi/mi15.png" width="23" height="18" alt="F_{2}">. The standard deviation of values obtained in the first
wavelet level <img id="Sx1.p3.m3" class="ltx_Math" style="vertical-align:-5px" src="mi/mi14.png" width="23" height="18" alt="F_{1}"> can be used as a molecular detection <a href="../../detectors/Threshold.html" title="" class="ltx_ref">threshold</a>
in the filtered image <cite class="ltx_cite">[<a href="#bib.bib16" title="Wavelet analysis for single molecule localization microscopy" class="ltx_ref">1</a>]</cite>.</p>
</div>
<div id="Sx1.p4" class="ltx_para">
<p class="ltx_p">The wavelet filter typically uses a <a href="Convolution.html" title="" class="ltx_ref">convolution kernel</a>
based on normalized B-spline basis functions <cite class="ltx_cite">[<a href="#bib.bib35" title="Astronomical Image and Data Analysis" class="ltx_ref">2</a>, <a href="#bib.bib16" title="Wavelet analysis for single molecule localization microscopy" class="ltx_ref">1</a>]</cite>.
In our case, the B-spline basis function of order <img id="Sx1.p4.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi30.png" width="59" height="21" alt="q\in\mathbb{N}^{+}">
in the variable <img id="Sx1.p4.m2" class="ltx_Math" style="vertical-align:-3px" src="mi/mi34.png" width="45" height="17" alt="t\in\mathbb{R}">, written as <img id="Sx1.p4.m3" class="ltx_Math" style="vertical-align:-7px" src="mi/mi13.png" width="48" height="22" alt="B_{q}\left(t\right)">,
is defined recursively as follows. For <img id="Sx1.p4.m4" class="ltx_Math" style="vertical-align:-5px" src="mi/mi27.png" width="45" height="19" alt="q=1">,</p>
<table id="Sx1.Ex1" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.Ex1.m1" class="ltx_Math" style="vertical-align:-24px" src="mi/mi8.png" width="227" height="57" alt="B_{1}\left(t\right)=\begin{cases}1&amp;\quad\mathrm{if}\;0\leq t&lt;1\,,\\
0&amp;\quad\mathrm{otherwise}\,,\end{cases}"></td>
<td class="ltx_eqn_pad"></td>
</tr>
</table>
<p class="ltx_p">and for <img id="Sx1.p4.m5" class="ltx_Math" style="vertical-align:-5px" src="mi/mi28.png" width="45" height="19" alt="q&gt;1">,</p>
<table id="Sx1.Ex2" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.Ex2.m1" class="ltx_Math" style="vertical-align:-18px" src="mi/mi9.png" width="341" height="42" alt="B_{q}\left(t\right)=\frac{t}{q-1}B_{q-1}\left(t\right)+\frac{q-t}{q-1}B_{q-1}%
\left(t-1\right)."></td>
<td class="ltx_eqn_pad"></td>
</tr>
</table>
<p class="ltx_p">Here <img id="Sx1.p4.m6" class="ltx_Math" style="vertical-align:-7px" src="mi/mi13.png" width="48" height="22" alt="B_{q}\left(t\right)"> has non-zero values only in the interval
<img id="Sx1.p4.m7" class="ltx_Math" style="vertical-align:-5px" src="mi/mi12.png" width="75" height="19" alt="0\leq t&lt;q">.</p>
</div>
<div id="Sx1.p5" class="ltx_para">
<p class="ltx_p">The first wavelet level <img id="Sx1.p5.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi14.png" width="23" height="18" alt="F_{1}"> is computed using the convolution
kernel <img id="Sx1.p5.m2" class="ltx_Math" style="vertical-align:-7px" src="mi/mi19.png" width="149" height="22" alt="\boldsymbol{k}_{1}=\left[k_{1,1},\ldots,k_{1,l}\right]">,
where <img id="Sx1.p5.m3" class="ltx_Math" style="vertical-align:-8px" src="mi/mi25.png" width="214" height="25" alt="k_{1,i}\left(x\mid q,s\right)=aB_{q}\left(\frac{x}{s}+\frac{q}{2}\right)">,
<img id="Sx1.p5.m4" class="ltx_Math" style="vertical-align:-5px" src="mi/mi4.png" width="87" height="19" alt="i=1,\ldots,l">, <img id="Sx1.p5.m5" class="ltx_Math" style="vertical-align:-6px" src="mi/mi35.png" width="135" height="21" alt="x=i-\left(l+1\right)/2">, and <img id="Sx1.p5.m6" class="ltx_Math" style="vertical-align:-8px" src="mi/mi26.png" width="110" height="25" alt="l=2\left\lceil\frac{qs}{2}\right\rceil-1">.
Here <img id="Sx1.p5.m7" class="ltx_Math" style="vertical-align:-3px" src="mi/mi32.png" width="45" height="16" alt="s&gt;0"> defines a scaling factor of the variable <img id="Sx1.p5.m8" class="ltx_Math" style="vertical-align:-2px" src="mi/mi36.png" width="14" height="12" alt="x">, and <img id="Sx1.p5.m9" class="ltx_Math" style="vertical-align:-2px" src="mi/mi23.png" width="13" height="12" alt="a">
is a scaling factor such that <img id="Sx1.p5.m10" class="ltx_Math" style="vertical-align:-7px" src="mi/mi22.png" width="90" height="22" alt="\sum_{i}k_{1,i}=1">. Kernels for higher
wavelet levels are obtained by inserting zeros between the kernel
coefficients. Users need to input the order <img id="Sx1.p5.m11" class="ltx_Math" style="vertical-align:-5px" src="mi/mi29.png" width="13" height="15" alt="q"> of the B-spline basis
function and the scaling factor <img id="Sx1.p5.m12" class="ltx_Math" style="vertical-align:-2px" src="mi/mi33.png" width="12" height="12" alt="s">.</p>
</div>
<div id="Sx1.p6" class="ltx_para">
<p class="ltx_p">In our implementation, the default kernel uses a B-spline basis function
of the third order with a scaling factor <img id="Sx1.p6.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi31.png" width="45" height="16" alt="s=2">. These settings yield
kernels <img id="Sx1.p6.m2" class="ltx_Math" style="vertical-align:-8px" src="mi/mi18.png" width="174" height="29" alt="\boldsymbol{k}_{1}=\left[\frac{1}{16},\frac{1}{4},\frac{3}{8},\frac{1}{4},%
\frac{1}{16}\right]^{\top}">
and <img id="Sx1.p6.m3" class="ltx_Math" style="vertical-align:-8px" src="mi/mi20.png" width="241" height="29" alt="\boldsymbol{k}_{2}=\left[\frac{1}{16},0,\frac{1}{4},0,\frac{3}{8},0,\frac{1}{4%
},0,\frac{1}{16}\right]^{\top}">,
as suggested in <cite class="ltx_cite">[<a href="#bib.bib35" title="Astronomical Image and Data Analysis" class="ltx_ref">2</a>, <a href="#bib.bib16" title="Wavelet analysis for single molecule localization microscopy" class="ltx_ref">1</a>]</cite>.</p>
</div>
<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Threshold for approximate localization of molecules</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<p class="ltx_p">The <a href="../../detectors/Threshold.html" title="" class="ltx_ref">threshold</a> value can be
specified by users as an expression combining mathematical functions
and operators with variables based on the current raw or filtered
image. Variables provided by this filter are:</p>
</div>
<div id="Sx1.SSx1.p2" class="ltx_para">
<table class="ltx_tabular ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">Wave.I</span></th>
<th class="ltx_td ltx_align_left">current raw input image</th>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">Wave.F</span></th>
<th class="ltx_td ltx_align_left">corresponding filtered image</th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">Wave.F1</span></th>
<td class="ltx_td ltx_align_left">first wavelet level <img id="Sx1.SSx1.p2.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi14.png" width="23" height="18" alt="F_{1}"> of the current input image</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">Wave.F2</span></th>
<td class="ltx_td ltx_align_left">second wavelet level <img id="Sx1.SSx1.p2.m2" class="ltx_Math" style="vertical-align:-5px" src="mi/mi15.png" width="23" height="18" alt="F_{2}"> of the current input image</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="Sx1.SSx2" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx2.p1" class="ltx_para">
<ul id="I1" class="ltx_itemize">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="Filters.html" title="" class="ltx_ref">Image filtering and feature enhancement</a></p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="../../detectors/Detectors.html" title="" class="ltx_ref">Finding approximate positions of molecules</a></p>
</div>
</li>
</ul>
</div>
</div>
</div>
<div id="bib" class="ltx_bibliography">
<h1 class="ltx_title ltx_title_bibliography">References</h1>

<ul id="L1" class="ltx_biblist">
<li id="bib.bib16" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[1]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">I. Izeddin, J. Boulanger, V. Racine, C. G. Specht, A. Kechkar, D. Nair, A. Triller, D. Choquet, M. Dahan and J. B. Sibarita</span><span class="ltx_text ltx_bib_year">(2012)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Wavelet analysis for single molecule localization microscopy</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Optics Express</span> <span class="ltx_text ltx_bib_volume">20</span> (<span class="ltx_text ltx_bib_number">3</span>), <span class="ltx_text ltx_bib_pages"> pp. 2081–95</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1364/OE.20.002081" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="Wavelet filter" class="ltx_ref"><span class="ltx_text ltx_ref_title">Wavelet filter</span></a>,
<a href="#Sx1.p3" title="Wavelet filter" class="ltx_ref"><span class="ltx_text ltx_ref_title">Wavelet filter</span></a>,
<a href="#Sx1.p4" title="Wavelet filter" class="ltx_ref"><span class="ltx_text ltx_ref_title">Wavelet filter</span></a>,
<a href="#Sx1.p6" title="Wavelet filter" class="ltx_ref"><span class="ltx_text ltx_ref_title">Wavelet filter</span></a>.
</span>
</li>
<li id="bib.bib35" class="ltx_bibitem ltx_bib_book">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[2]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">J.-L. Starck and F. Murtagh</span><span class="ltx_text ltx_bib_year">(2002)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Astronomical Image and Data Analysis</span>,
</span>
<span class="ltx_bibblock"> <span class="ltx_text ltx_bib_publisher">Springer-Verlag</span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="Wavelet filter" class="ltx_ref"><span class="ltx_text ltx_ref_title">Wavelet filter</span></a>,
<a href="#Sx1.p4" title="Wavelet filter" class="ltx_ref"><span class="ltx_text ltx_ref_title">Wavelet filter</span></a>,
<a href="#Sx1.p6" title="Wavelet filter" class="ltx_ref"><span class="ltx_text ltx_ref_title">Wavelet filter</span></a>.
</span>
</li>
</ul>
</div>
</div>
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